Global convergence of sequential injective algorithm for weakly univalent vector equation: application to regularized smoothing Newton algorithm

It is known that the complementarity problems and the variational inequality problems are reformulated equivalently as a vector equation by using the natural residual or Fischer-Burmeister function. In this short paper, we first study the global convergence of a sequential injective algorithm for weakly univalent vector equation. Then, we apply the convergence analysis to the regularized smoothing Newton algorithm for mixed nonlinear second-order cone complementarity problems. We prove the global convergence property under the (Cartesian) P0 assumption, which is strictly weaker than the original monotonicity assumption. 1 Sequential injective algorithm for weakly univalent equation Consider the following vector equation (VE): H(z) = 0, (1.1) where H : D ⊆ Rn → Rn is continuous over the domain D ⊆ Rn, but need not be linear or differentiable. Notice that many classes of problems such as linear complementarity problem (LCP), nonlinear complementarity problem (NCP), second-order cone complementarity problem (SOCCP) [3, 4], symmetric cone complementarity problem (SCCP) [5], variational inequality problem (VIP) and fixed point problem can be cast as VE (1.1). (For more details, see [2]). In this section, we first study the global convergence of a certain conceptual algorithm for solving VE (1.1). Then, in the next section, we apply the obtained convergence theorem to the regularized smoothing Newton algorithm in a direct manner. 1.1 Weak univalence property In the convergence analysis, the notion of the weak univalence property plays an important role. Definition 1.1 (weak univalence property) [2, Sec. 3.6] Function H : D ⊆ Rn → Rn is said to be “weakly univalent” if it is continuous and there exists a sequence of continuous and injective functions {H̃k} converging to H uniformly∗1 over any bounded subset of D. We can easily see that any weakly univalent function is continuous. Moreover, any P0 function or monotone function is weakly univalent. For VE (1.1), we suppose that H satisfies the following assumption: ∗Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan (s [email protected]. tohoku.ac.jp) ∗1We say that the sequence of functions {H̃k} converges toH uniformly over the bounded set Ω, if sup{‖H̃k(w)−H(w)‖ : w ∈ Ω} converges to 0 as k → ∞.